Perform the row operation, $R_3+R_2\rightarrow R_3$, on the following matrix. $\left[\begin{array} {ccc} -3 & 0 & 1 & 9 \\ 8 & 9 & -7 & -7 \\ -9 & 19 & 1 & 13 \end{array} \right] $
Solution: Background There are three basic row operations that can be performed on matrices. $R_i \leftrightarrow R_j$. This symbol tells us to interchange rows $i$ and $j$. $cR_i \rightarrow R_i$. This symbol tells us to multiply a row $i$ by a constant $c$. $R_i + cR_j \rightarrow R_i$. This symbol tells us to add $c$ times row $j$ to row $i$. Finding the new row to be used For the given matrix, $R_2$ and $R_3$ are given below. $R_2=\left[\begin{array} {ccc} 8 & 9 & -7 & -7 \end{array} \right] ~~~~~ R_3=\left[\begin{array} {ccc} -9 & 19 & 1 & 13 \end{array} \right]$ We are asked to perform the row operation, $R_3+R_2\rightarrow R_3$. Therefore, we must add $R_2$ to $R_3$. $\begin{aligned}R_3+R_2 &= \left[\begin{array} {ccc} -9 & 19 & 1 & 13 \end{array} \right] + \left[\begin{array} {ccc} 8 & 9 & -7 & -7 \end{array} \right] \\\\&=\left[\begin{array} {ccc} -1 & 28 & -6 & 6 \end{array} \right]\end{aligned}$ Substituting the row Now, we must substitute row $R_3$ with $R_3+R_2$. $\left[\begin{array} {ccc} -3 & 0 & 1 & 9 \\ 8 & 9 & -7 & -7 \\ {-9} & {19} & {1} & {13} \end{array} \right] \xrightarrow{R_3+R_2\rightarrow R_3} \left[\begin{array} {ccc} -3 & 0 & 1 & 9 \\ 8 & 9 & -7 & -7 \\ {-1} & {28} & {-6} & {6} \end{array} \right]$ Summary Our resultant matrix is the following. $\left[\begin{array} {ccc} -3 & 0 & 1 & 9 \\ 8 & 9 & -7 & -7 \\ -1 & 28 & -6 & 6 \end{array} \right]$